Transcript

Robyn Williams: Do you know about Emilie du Chatelet? She was a brilliant mathematician and physicist who translated Isaac Newtown's work into French and corrected some of his misapprehensions as she went. A close friend of Voltaire's, she had a profound effect on intellectual life in France, but lost her own through giving birth when just 40. The parallels with her modern equivalent are haunting. This one came from Iran and her poignant story is told here by Nalini Joshi, Professor of Maths at the University of Sydney.

Nalini Joshi: On 16 June 2014, I received an email from the President of the International Mathematical Union. The subject of the email was a question “Will you be at ICM?” and the body of the email consisted of just two lines: “and will you have some disposable time? I have a favor to ask ... Best, Ingrid”. The sender was Professor Ingrid Daubechies and ICM stands for the International Mathematical Congress, which is held every 4 years. The most anticipated event at each ICM is the award of the Fields medal, arguably the most celebrated prize in mathematics. In 2014, the ICM was to be held in Seoul, Korea.

When I answered Ingrid’s email, I learnt that I was to be part of a small group of female mathematicians to be entrusted with a special job in Seoul, involving such amazing news that it set a bell ringing in my heart, a bell that is still tinkling today. This group, whose nickname was the MM shield, learnt that a Fields medal was to be awarded to a female mathematician for the first time in its long and luminous history. We were asked to be on call to provide support and help for the recipient at ICM.

To understand why this was such momentous news, you have to know something about the Fields medal. The prestigious medal, often called the Nobel prize for mathematics, is awarded to recognize extraordinary results in mathematics by a mathematician under the age of 40. Before that news arrived in my inbox, 52 Fields medals had been awarded, all to men. The bell in my heart was my emotional reaction to the news. The established ranks of the mathematical world were going to acknowledge not only that women can do maths, but that their achievements can be as brilliant as the highest achievements ever recognized in recorded mathematical history.

One of the four brilliant mathematicians who were awarded Fields medals at the 2014 ICM was Maryam Mirzakhani, who was born and educated in Iran, and had completed a PhD in mathematics at Harvard University. Behind that brief outline, there was an exceptional person with an astonishing history. Iran admitted girls to their International Mathematics Olympiad (IMO) team for the very first time in 1994. The contest is difficult, involving 6 problems worth 7 marks each, distributed over two consecutive days. Maryam was a member of that first team to include girls. Maryam did not just participate, her performance was spectacular: she dropped only 1 mark in 1994, and in the next year, she achieved a perfect score. She was the first Iranian ever to have been awarded two gold medals and a perfect score in the IMO. An observer who witnessed the IMO medal ceremonies in 1994 and 1995 wrote to me that: “The two girls in the Iranian team were completely covered in black - except for their feet. When the Iranians went to collect their medals, one had to infer what was underneath from the confident way the girls' trainers strode visibly across the stage. It was very, very impressive - if a rather unusual way to witness/infer youthful ambition.”

Much later, that drive and ambition led Maryam to move to the US to pursue graduate study in Mathematics at Harvard University, where her PhD advisor, Curtis McMullen, also a Fields medallist, said she showed “determination and relentless questioning”. One of my mathematical heroes, Sofia Kowalevskaya said “Many who have had an opportunity of knowing … more about mathematics confuse it with arithmetic, and consider it an arid science. In reality, however, it … requires a great deal of imagination.” In my biased and humble opinion, mathematics is actually one of the most creative and richest of human inventions.

To sustain the great collective progress of associations and ideas, mathematicians have developed many subfields and technical specialisations. Many research mathematicians spend all of their working lives as highly accomplished technical specialists working within one subfield of mathematics, much like a performer might interpret and play glorious music on just one instrument. Maryam’s best results were breathtaking because they transcended the conventional boundaries of mathematics and provided unexpected insights into other areas.

To give an example, imagine wrapping a ribbon around a doughnut. You could put the ribbon around a longitudinal cross-section or you could put it around its equator, or combinations of these. You might be able to shuffle the ribbon around, but you can’t shorten the length of a ribbon without it falling off or cutting into the doughnut. The closed curve represented by each ribbon is called a geodesic. On Earth the geodesics are given by great circles. Imagine you can describe geodesics even as a surface deforms continuously into other shapes, as for example a doughnut becomes a coffee mug. Maryam studied geodesics on hyperbolic surfaces, which are surfaces that are shaped like saddles. Can we estimate the number of geodesics on each type of hyperbolic surface as their length grows?

In her PhD thesis, Maryam proved a sharp estimate of this number on hyperbolic spaces by studying all possible deformations of a surface. One of the stunning results that came out of her work was a proof of a famous result in an entirely different setting: that is, a proof of Witten’s conjecture, which concerned the physics of two different models of 2-dimensional quantum gravity in string theory. But she didn’t stop there. For example, consider what happens when you cut and slip surfaces along geodesics, and glue them back together over and over again. These are like earthquakes occurring on that surface and about 3 decades earlier another Fields medallist, William Thurston, had described a system called “earthquake flow” that arose from such constructions. Maryam proved that the dynamics of earthquake flow is chaotic. This result blazed a new path into an area that was previously thought to be completely obscure.

To study such problems, Maryam used the mathematical setting of moduli spaces, which has always been regarded as tricky and mysterious. For example, suppose we draw a family of parabolas that change continuously as we change a parameter. These are zero sets of quadratics with coefficients that change continuously. More generally, we could consider a family of curves given as the zero sets of polynomials containing parameters. When we study these curves from the point of view of the space of parameters, rather than just curves, we get a moduli space.

It is unusual for a mathematician to develop a workable intuition in the setting of moduli space. It was amazing that Maryam could delve not only into its mysteries, but actually pull out deeply held treasures she discovered in such spaces. Each time she proved something, it created ramifications that travelled far and wide into other areas.

Another example of this arose in her recent work on complex geodesics. No one expected to see anything but complicated, untameable wilderness when these are studied. But it turns out that their closures in moduli space are much simpler than anyone anticipated. These powerful results and their surprising regularity are likely to lead to major discoveries in many other directions. I met Maryam and her husband Jan and their little daughter Anahita at the 2014 ICM in Seoul. Maryam was a devoted mother, always holding Anahita when she was drawn into photographs near her, and paying close attention to Anahita's reactions and emotions.

The year prior to the award of the Fields medal, Maryam had been diagnosed with breast cancer. The MM shield was formed to give her support and protection from possibly excessive media intrusions and to allow her the space and time to rest. We were delighted that she had recovered enough to attend the ICM and accept her Fields medal, although she could not stay to deliver her invited lecture. So it was a shock to hear the news on the 14 July, 2017, that Maryam Mirzakhani died. As a female mathematician myself, I felt like a close family member passed away. It was like the universe had lost an avatar, a rare incarnation in the universe of all possibilities.

But I know that in the world somewhere, now or in the future, there is a little girl or a woman, brave, persistent, driven to discovery through relentless questioning, who is deeply talented, who may win a Fields medal again. The bell is still ringing in my heart.

Robyn Williams: And many of us share your feeling. Nalini Joshi is a professor of maths at the University of Sydney. Next week - humility. Is it an essential ingredient for leadership? We go to Canberra to find out. I'm Robyn Williams.

Guests

Nalini Joshi

Australian Research Council Georgina Sweet Australian Laureate Fellow in Mathematics, University of Sydney.